Perturbations of flows of incompressible nonlinearly viscous and viscoplastic fluids caused by variations in material functions

Material functions are necessary element of the constitutive relations determining any model of continuum. These functions can be defined as a collection of objects from which the operator of constitutive relations can be reconstructed completely. The material functions are found in test experiments and show the differences between a given medium and other media in the framework of the same model [1]. The “test experiment theory” is an important part of modern experimental mechanics.Just as in any experiment, from determining the viscosity coefficient by using the rotational viscosimeters to constructing the yield surface by using machines combined loading, the material functions are determined with an unavoidable error. For example, experimenters know that, in experiments with arbitrary accuracy, the moduli of elasticity can only be measured with an unimprovable tolerance of about 7%. Starting already from [2], the investigators’ attention has been repeatedly drawn to the fact that it is necessary to take into account this tolerance in determining the material constants, functions, and functionals in problems of mechanics and especially in analyzing the stability of deformation processes. Mathematically, this means that problems of stability under perturbations of the initial data, external constantly acting forces, domain boundaries, etc. should be supplemented with the assumption that the material functions have unknown perturbations of a certain class [3].The variations of material functions in the framework of the linearized stability theory were considered in [2, 4, 5]. In what follows, we study isotropic tensor functions in the most general case of scalar and tensor nonlinearity. These functions are assigned the meaning of constitutive relations between the stress and strain rate tensors in continuum. These constitutive relations contain scalar material functions of invariants on which, as follows from the above, some variations proportional to a small physical parameter α can be imposed. These variations imply perturbations of the tensor function itself. The components of such perturbations linear and quadratic in α are determined. In each of the approximations, we write out a closed system of equations consisting of the equations of motion (linear in the variables of the respective approximation) and the incompressibility condition.We analyze tensor-linear functions with arbitrary scalar rheology inmore detail. Materials with such constitutive relations include non-Newtonian viscous fluids and viscoplastic materials. Viscoplastic materials are characterized by the existence of rigidity zones, where the stress intensity is less than the yield strength. We derive equations for the boundaries of the rigidity zones in the perturbed motion, in particular, for the case in which the unperturbed medium is a viscous Newtonian fluid. Throughout the paper, index-free notation is used.     

Non-linear shear vibrations of piezoceramic actuators

A wide range of non-linear effects are observed in piezoceramic materials. For small stresses and weak electric fields, piezoceramics are usually described by linearized constitutive equations around an operating point. However, typical non-linear vibration behavior is observed at weak electric fields near resonance frequency excitations of the piezoceramics. This non-linear behavior is observed in terms of a softening behavior and the decrease of normalized amplitude response with increase in excitation voltage. In this paper the authors have attempted to model this behavior using higher order cubic conservative and non-conservative terms in the constitutive equations. Two-dimensional kinematic relations are assumed, which satisfy the considered reduced set of constitutive relations. Hamilton's principle for the piezoelectric material is applied to obtain the non-linear equation of motion of the piezoceramic rectangular parallelepiped specimen, and the Ritz method is used to discretize it. The resulting equation of motion is solved using a perturbation technique. Linear and non-linear parameters for the model are identified. The results from the theoretical model and the experiments are compared. The non-linear effects described in this paper may have strong influence on the design of the devices, e.g. ultrasonic motors, which utilize the piezoceramics near the resonance frequency excitation.     

Thermal Instability in a Porous Medium Layer Saturated with a Viscoelastic Nanofluid

The onset of convection in a horizontal layer of a porous medium saturated with a viscoelastic nanofluid was studied in this article. The modified Darcy model was applied to simulate the momentum equation in porous media. An Oldroyd-B type constitutive equation was used to describe the rheological behavior of viscoelastic nanofluids. The model used for the viscoelastic nanofluid incorporates the effects of Brownian motion and thermophoresis. The onset criterion for stationary and oscillatory convection was analytically derived. The effects of the concentration Rayleigh number, Prandtl number, Lewis number, capacity ratio, relaxation, and retardation parameters on the stability of the system were investigated. Oscillatory instability is possible in both bottom- and top-heavy nanoparticle distributions. Results indicated that there is competition among the processes of thermophoresis, Brownian diffusion, and viscoelasticity that causes the convection to set in through oscillatory rather than stationary modes. Regimes of stationary and oscillatory convection for various parameters were derived and are discussed in detail.     

Crystal microstructure and rheology of highly-concentrated ferromagnetic suspensions

For a highly-concentrated suspension of ferromagnetic particles characterized by dipolar interaction, when Brownian motion can be neglected, a model of a viscoelastic magnetizable continuum is developed. The constitutive equations and the kinetic coefficients are determined in the lubrication approximation on the basis of the periodic array model. In the problem of slow medium deformation in a constant magnetic field, for certain values of the determining parameters, an increase in the effective viscosity with decrease in the field strength due to the strong dipolar interaction of the particles is detected.     

关于颗粒悬浮机理和悬浮动的讨论

从分析气体分子的悬浮和静水中Ｂｒｏｗｎ微粒的悬浮之机理出发,论述了重力场中粒子（分子、微粒等）的悬浮不一定需要其它外力,粒子本身的任何形式的无规则运动,达到一定强度后都能使粒子弥散悬浮．河流中的泥沙颗粒和气（水）力输送管道中的颗粒的悬浮也主要靠颗粒物的无规则运动．作用于颗粒的升力和其它力可改变颗粒悬浮沿高度的分布,但仅用这些力（若无任何无规则运动）无法解释颗粒的弥散悬浮状态．讨论了颗粒对流动阻力的双重作用：支持颗粒悬浮的湍流脉动因引入颗粒而削弱,这是颗粒的减阻作用;颗粒增阻的一个主要机制是,流体给予颗粒的水平动量在颗粒一壁面碰撞中不断地损失．用悬浮动概念解释颗粒引起的增阻是不正确的．     

On the influence of near-wall forces in particle-laden channel flows

Computation of a turbulent dilute gas–solid channel flow has been undertaken to study the influence of using wall-corrected drag coefficients and of the lift force on the dispersed phase characteristics. The incompressible Navier–Stokes equations governing the carrier flow were solved by using a direct numerical simulation approach and coupled with a Lagrangian particle tracking. Calculations were performed at Reynolds number based on the wall-shear velocity and channel half-width, Re_τ ≈ 184, and for three different sets of solid particles. For each particle set, two cases were examined, in the first one the particle motion was governed by both drag and lift wall-corrected forces, whereas in the other one, the standard drag force (not corrected) was solely acting. The lift force model used represents the most accurate available expression since it accounts for weak and strong shear as well as for wall effects. For this study, we considered elastic collisions for particles contacting the walls and that no external forces were acting. Present results indicate that the use of the lift force and of the drag corrections does not lead to significant changes in the statistical properties of the solid phase, at the exception of some statistics for the high inertia particles.     

Models and problems for saturated porous media

A multiphase model of a liquid- and gas-saturated porous medium is proposed. The model takes into account the finite deformations of a skeleton, arbitrary flows of liquids and gases, and the phase mass transfer between the skeleton and the liquids. Some basic relations are given for the corresponding boundary value problems in the cases of arbitrary and small motions. The constitutive relations describing the properties of the skeleton resistance to deformation (skeleton stresses) and the mutual resistance of components (internal interaction) are considered in detail. The internal actions exerted on the skeleton by a moving liquid (gas) are discussed when these actions may take the form of drag forces, lift (displacing) forces, overturning moments, and rotational (screw) moments.     

A new constitutive model for monodispersed suspensions of spheres at high concentrations

We describe a constitutive model for monodispersed concentrated suspensions of spheres at high volume fractions. In this model the motion of a pair of generic spheres is represented by that of a pair of force-free and torque-free spheres tumbling along with the flow. The interaction with the surrounding spheres is modelled by an anisotropic diffusion process. Viscometric and elongational properties of the model are obtained by a Brownian dynamic simulation. Received: 24 September 1998 Accepted: 1 December 1998     

On flows of granular materials

This article reviews the behavior of materials made up of a large assemblage of solid particles under rapid and quasi static deformations. The focus is on flows at relatively high concentrations and for conditions when the interstitial fluid plays an insignificant role. The momentum and energy exchange processes are then primarily governed by interparticle collisions and Coulomb-type frictional contact. We first discuss some physical behavior —dilatancy, internal friction, fluidization and particle segregation — that are typical to the understanding of granular flows. Bagnold's seminal Couette flow experiments and his simple stress analysis are then used to motivate the first constitutive theories that use a microstructural variable — the fluctuation energy or granular temperature — governing the subscale fluctuating motion. The kinetic theories formalize the derivation of the field equations of bulk mass, momentum and energy, and permit derivation of constitutive relations for stress, flux of fluctuation energy and its dissipation rate for simple particle assemblages and when frictional rubbing contact can be ignored. These statistical considerations also show that formulation of boundary conditions needs special attention. The frictional-collisional constitutive behavior in which both Coulomb-type rubbing contact and collisional encounters are significant are discussed. There is as yet no rigorous formulation. We finally present a phenomenological approach that describes rapid flows of granular materials under simultaneous transport of heat and close with a summary of stability analyses of the basic flow down an inclined plane.Dedicated to Professor Dr.-Ing. Franz Gustav Kollmann on the occasion of his sixtieth brithday     

Coupled nonlinear constitutive models for rarefied and microscale gas flows: subtle interplay of kinematics and dissipation effects

The constitutive relations of gases in a thermal nonequilibrium (rarefied and microscale) can be derived by applying the moment method to the Boltzmann equation. In this work, a model constitutive relation determined on the basis of the moment method is developed and applied to some challenging problems in which classical hydrodynamic theories including the Navier–Stokes–Fourier theory are shown to predict qualitatively wrong results. Analysis of coupled nonlinear constitutive models enables the fundamentals of gas flows in thermal nonequilibrium to be identified: namely, nonlinear, asymmetric, and coupled relations between stresses and the shear rate; and effect of the bulk viscosity. In addition, the new theory explains the central minimum of the temperature profile in a force-driven Poiseuille gas flow, which is a well-known problem that renders the classical hydrodynamic theory a global failure.     

The Effect of Vertical Throughflow on Thermal Instability in a Porous Medium Layer Saturated by a Nanofluid

The effect of vertical throughflow on the onset of convection in a horizontal layer of a porous medium saturated by a nanofluid is studied analytically. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The dependences of the critical Rayleigh number for the non-oscillatory and oscillatory modes of instability on the thermophoresis and Brownian motion parameters for the cases with and without throughflow are investigated.     

Magnetohydrodynamic and heat transfer effects on the stagnation-point flow of an electrically conducting nanofluid past a porous vertical shrinking/stretching sheet in the presence of variable stream conditions

A steady two-dimensional magnetohydrodynamic stagnation-point flow of an electrically conducting fluid and heat transfer with thermal radiation of a nanofluid past a shrinking and stretching sheet is investigated numerically. The model used for the nanofluid incorporates the effects of the Brownian motion and thermophoresis. A similarity transformation is used to convert the governing nonlinear boundary-layer equations into coupled higher-order nonlinear ordinary differential equations. The result shows that the velocity, temperature, and concentration profiles are significantly influenced by the Brownian motion, heat radiation, and thermophoresis particle deposition.     

Parametric excitations and lock-in of flexible hydrofoils in two-phase flows

This work introduces a reduced-order method to study the parametric excitations and lock-in of flexible hydrofoils caused by unsteady two-phase (cavitating) flow. The reduced-order method is based on a 1-DOF structural model coupled with a van der Pol wake oscillator with empirically derived relations for the variation in lift, cavity-length, and cavity-shedding frequency as a function of a non-dimensional cavitation parameter. The results are compared with several available data from both numerical simulations and experimental measurements. The frequency content of both the predicted and measured vibrations suggested that, in addition to the primary cavity-shedding frequency and the hydrofoil natural frequencies, unsteady two-phase flows may excite additional modulated frequencies due to time-varying fluid-added mass effects. The results show that these frequency modulations might cause the flexible hydrofoil to undergo higher-order resonances, as well as parametric resonances. While the maximum deformations for the primary and higher-order resonances were observed to damp out, parametric resonances might persist even with realistic fluid damping coefficients (4–12%). It was observed that with higher effective foil flexibility, the cavity-shedding frequencies may be significantly modified from the rigid foil trends, and may instead lock-in with the system natural frequencies.     

Analytical and numerical solutions for axisymmetric flow of nanofluid due to non-linearly stretching sheet

This article reports the laminar axisymmetric flow of nanofluid over a non-linearly stretching sheet. The model used for nanofluid contains the simultaneous effects of Brownian motion and thermophoretic diffusion of nanoparticles. The recently proposed boundary condition is considered which requires the mass flux of nanoparticles at the wall to be zero. Analytic solutions of the arising boundary value problem are obtained by optimal homotopy analysis method. Moreover the numerical solutions are computed by Keller–Box method. Both the solutions are found in excellent agreement. The behavior of Brownian motion on the fluid temperature and wall heat transfer rate is insignificant. Further the nanoparticle volume fraction distribution is found to be negative near the vicinity of the stretching sheet.     

The onset of convection in a horizontal nanofluid layer of finite depth

This paper presents a linear stability analysis for the onset of natural convection in a horizontal nanofluid layer. The employed model incorporates the effects of Brownian motion and thermophoresis. Both monotonic and oscillatory convection for free–free, rigid–rigid, and rigid–free boundaries are investigated. The oscillatory instability is possible when nanoparticles concentrate near the bottom of the layer, so that the density gradient caused by such a bottom-heavy nanoparticle distribution competes with the density variation caused by heating from the bottom. It is established that the instability is almost purely a phenomenon due to buoyancy coupled with the conservation of nanoparticles. It is independent of the contributions of Brownian motion and thermophoresis to the thermal energy equation. Rather, the Brownian motion and thermophoresis enter to produce their effects directly into the equation expressing the conservation of nanoparticles so that the temperature and the particle density are coupled in a particular way, and that results in the thermal and concentration buoyancy effects being coupled in the same way.     

Dynamics and rheology of a dilute suspension of dipolar nonspherical particles in an external field: Part 1. Steady shear flows

Suspensions of small nonspherical particles having dipolar moments exhibit non-Newtonian behavior under the influence of shear and external fields. Numerical methods are presented for calculating the rheological and rheo-optical properties of dilute suspensions of Brownian particles having permanent dipoles subject to time-dependent shear and external fields. The numerical methods employ the Galerkin method of weighted residuals to solve the differential equation for the particle orientation distribution function. The steady-state shear flow intrinsic viscosity of suspensions of particles with sufficiently extreme aspect ratio is predicted to exhibit a maximum value attained for intermediate shear rates at selected field orientations. These numerical results provide valuable insight into the coupling which occurs between the effects of rotary Brownian motion, the hydrodynamic resistance of nonspherical particles, and the external torque exerted on dipolar particles. The results are applicable to both suspensions of magnetic particles and electrically dipolar particles.     

关于颗粒悬浮机理和悬浮动的讨论

从分析气体分子的悬浮和静水中Ｂｒｏｗｎ微粒的悬浮之机理出发，论述了重力场中粒子（分子、微粒等）的悬浮不一定需要其它外力，粒子本身的任何形式的无规则运动，达到一定强度后都能使粒子弥散悬浮．河流中的泥沙颗粒和气（水）力输送管道中的颗粒的悬浮也主要靠颗粒物的无规则运动．作用于颗粒的升力和其它力可改变颗粒悬浮沿高度的分布，但仅用这些力（若无任何无规则运动）无法解释颗粒的弥散悬浮状态．讨论了颗粒对流动阻力的双重作用：支持颗粒悬浮的湍流脉动因引入颗粒而削弱，这是颗粒的减阻作用；颗粒增阻的一个主要机制是，流体给予颗粒的水平动量在颗粒一壁面碰撞中不断地损失．用悬浮动概念解释颗粒引起的增阻是不正确的．     

Uniqueness of uniaxial elastic-plastic interface motions

The uniaxial motion of interfaces between regions deforming elastically and regions deforming plastically is considered. The governing constitutive, stress-rate/strain-rate equations in both elastic and plastic regions are taken to be non-linear. Discontinuity relations across such interfaces are established by the repeated differentiation of existing relations. The relations given by previous workers (especially R.J. Clifton, T.C.T. Ting, E.H. Lee and Th. von Kármán) are discussed. The precise situations in which they hold are considered, and it is shown that some of these relations, while apparently derived for different situations, can, in certain circumstances, be shown to be equivalent. It has been shown that six essentially different types of motion can occur, and, when the constitutive equations are linear, each type of motion is unique. This result is extended to the non-linear situation, by means of an established local expansion procedure. For the case of a meeting interaction of stress waves carrying initially linear profiles, the previous (linear) analysis given by L.W. Morland and A.D. Cox fails to distinguish between certain types of motion. This motion is reconsidered and it is shown how non-linearity in the constitutive laws serves to determine uniquely the type of motion that takes place.